On finite groups containing three 𝐶𝐶-subgroups

Arad, Zvi; Ferguson, Pamela

doi:10.1090/S0002-9939-1980-0574503-0

On finite groups containing three $CC$-subgroups
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by Zvi Arad and Pamela Ferguson PDF

Proc. Amer. Math. Soc. 80 (1980), 27-33 Request permission

Abstract:

A finite group G has a self-centralization system of type $(2|{A_1}|,4|{A_2}|,4|{A_3}|)$ if G contains three nonconjugate CC-subgroups ${A_1},{A_2},{A_3}$, such that $|{N_G}({A_1})| = 2|{A_1}|,|{N_G}({A_2})| = 4|{A_2}|,|{N_G}({A_3})| = 4|{A_3}|$. The authors prove that if a finite group G has a self-centralization system of type $(2|{A_1}|,4|{A_2}|,4|{A_3}|)$ and $|G| \leqslant 3|{A_1}{|^2}|{A_2}{|^2}|{A_3}{|^2}$, then G has a nilpotent normal subgroup N such that G/N is isomorphic to $Sz(q)$ for suitable q.

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